Olympiad problems

2007 Cuba MO, 4

Find all functions $f : R_+ \to R_+$ such that $$x^2(f(x)+f(y)) = (x+y)f(f(x)y)$$ for all positive real $x, y$.

2017 Czech And Slovak Olympiad III A, 3

Tags: functional equation , functional , algebra

Find all functions $f: R \to R$ such that for all real numbers $x, y$ holds $f(y - xy) = f(x)y + (x - 1)^2 f(y)$

1990 Romania Team Selection Test, 3

Tags: polynomial , algebra , functional , functional equation

Find all polynomials $P(x)$ such that $2P(2x^2 -1) = P(x)^2 -1$ for all $x$.

2018 Saudi Arabia IMO TST, 1

Tags: number theory , divide , divisible , functional

Find all functions $f : Z^+ \to Z^+$ satisfying $f (1) = 2, f (2) \ne 4$, and max $\{f (m) + f (n), m + n\} |$ min $\{2m + 2n, f (m + n) + 1\}$ for all $m, n \in Z^+$.

2004 Switzerland Team Selection Test, 11

Tags: functional equation , functional

Find all injective functions $f : R \to R$ such that for all real $x \ne y$ , $f\left(\frac{x+y}{x-y}\right) = \frac{f(x)+ f(y)}{f(x)- f(y)}$

2018 NZMOC Camp Selection Problems, 10

Tags: functional equation , functional , algebra

Find all functions $f : R \to R$ such that $$f(x)f(y) = f(xy + 1) + f(x - y) - 2$$ for all $x, y \in R$.

2018 Estonia Team Selection Test, 4

Tags: algebra , functional , functional equation

Find all functions $f : R \to R$ that satisfy $f (xy + f(xy)) = 2x f(y)$ for all $x, y \in R$

VI Soros Olympiad 1999 - 2000 (Russia), 10.3

Tags: algebra , functional equation , functional

Find all functions $f$ that map the set of real numbers into the set of real numbers, satisfying the following conditions: 1) $|f(x)|\ge 1$, 2) $f(x+y)=\frac{f(x)+f(y)}{1+f(x)f(y)}$ of all real values of $x $ and $y$.

1989 Swedish Mathematical Competition, 2

Tags: continuous , functional equation , functional , algebra

Find all continuous functions $f$ such that $f(x)+ f(x^2) = 0$ for all real numbers $x$.

2008 Cuba MO, 4

Tags: algebra , functional

Determine all functions $f : R \to R$ such that $f(xy + f(x)) =xf(y) + f(x)$ for all real numbers $x, y$.

VMEO I 2004, 5

Tags: functional equation , functional , algebra

Find all the functions $f:R \to R$ satisfying $$(x + y)(f (x)-f (y)) = f (x^2) - f (y^2),\, \forall x, y \in R$$

2007 Thailand Mathematical Olympiad, 9

Tags: functional equation , functional , algebra

Let $f : R \to R$ be a function satisfying the equation $f(x^2 + x + 3) + 2f(x^2 - 3x + 5) =6x^2 - 10x + 17$ for all real numbers $x$. What is the value of $f(85)$?

2006 Thailand Mathematical Olympiad, 6

Tags: algebra , functional equation , functional , trigonometry

A function $f : R \to R$ has $f(1) < 0$, and satisfy the functional equation $$f(\cos (x + y)) = (\cos x)f(\cos y) + 2f(\sin x)f(\sin y)$$ for all reals $x, y$. Compute $f \left(\frac{2006}{2549 }\right)$

VI Soros Olympiad 1999 - 2000 (Russia), 9.4

Tags: algebra , functional , functional inequation , inequalities

Is there a function $f(x)$, which satisfies both of the following conditions: a) if $x \ne y$, then $f(x)\ne f(y)$ b) for all real $x$, holds the inequality $f(x^2-1998x)-f^2(2x-1999)\ge \frac14$?

2009 Switzerland - Final Round, 6

Tags: functional , functional equation , algebra

Find all functions $f : R_{>0} \to R_{>0}$, which for all $x > y > z > 0$ is the following equation holds $$f(x - y + z) = f(x) + f(y) + f(z) - xy - yz + xz.$$

1996 Israel National Olympiad, 2

Tags: polynomial , algebra , functional equation , functional

Find all polynomials $P(x)$ satisfying $P(x+1)-2P(x)+P(x-1)= x$ for all $x$

1994 Italy TST, 3

Tags: functional equation , functional , algebra

Find all functions $f : R \to R$ satisfying the condition $f(x- f(y)) = 1+x-y$ for all $x,y \in R$.

1999 Switzerland Team Selection Test, 3

Tags: functional equation , functional , algebra

Find all functions $f : R -\{0\} \to R$ that satisfy $\frac{1}{x}f(-x)+ f\left(\frac{1}{x}\right)= x$ for all $x \ne 0$.

2015 Saudi Arabia Pre-TST, 2.2

Tags: algebra , functional

Find all functions $f : R \to R$ that satisfy $f(x + y^2 - f(y)) = f(x)$ for all $x,y \in R$. (Vo Quoc Ba Can)

2017 QEDMO 15th, 4

Tags: functional equation , functional , algebra

Find all functions $f: R \to R$ for which the image $f ([a, b])$ for all real $a \le b$ is (not necessarily closed!) interval of length $b - a$.

2018 Costa Rica - Final Round, F2

Tags: functional equation , functional , algebra , sequence , combinatorics

Consider $f (n, m)$ the number of finite sequences of $ 1$'s and $0$'s such that each sequence that starts at $0$, has exactly n $0$'s and $m$ $ 1$'s, and there are not three consecutive $0$'s or three $ 1$'s. Show that if $m, n> 1$, then $$f (n, m) = f (n-1, m-1) + f (n-1, m-2) + f (n-2, m-1) + f (n-2, m-2)$$

2000 All-Russian Olympiad Regional Round, 10.5

Tags: algebra , inequalities , function , functional , trigonometry

Is there a function $f(x)$ defined for all $x \in R$ and for all $x, y \in R $ satisfying the inequality $$|f(x + y) + \sin x + \sin y| < 2?$$

2019 Switzerland - Final Round, 6

Tags: algebra , functional , functional equation

Show that there exists no function $f : Z \to Z$ such that for all $m, n \in Z$ $$f(m + f(n)) = f(m) - n.$$

2019 Swedish Mathematical Competition, 5

Tags: algebra , inequalities , functional , functional inequality , functional equation

Let $f$ be a function that is defined for all positive integers and whose values are positive integers. For $f$ it also holds that $f (n + 1)> f (n)$ and $f (f (n)) = 3n$, for each positive integer $n$. Calculate $f (2019)$.

1995 Singapore Team Selection Test, 1

Tags: algebra , functional equation , functional

Let $N =\{1, 2, 3, ...\}$ be the set of all natural numbers and $f : N\to N$ be a function. Suppose $f(1) = 1$, $f(2n) = f(n)$ and $f(2n + 1) = f(2n) + 1$ for all natural numbers $n$. (i) Calculate the maximum value $M$ of $f(n)$ for $n \in N$ with $1 \le n \le 1994$. (ii) Find all $n \in N$, with 1 \le n \le 1994, such that $f(n) = M$.